An Explicit Parametrization of Closed Loops for Spatially Distributed Controllers with Sparsity Constraints

TL;DR

This paper introduces an explicit parametrization method for designing spatially sparse, distributed controllers in linear systems, enabling efficient synthesis and analysis of large-scale control architectures.

Contribution

It provides a new explicit parameterization of all stabilized closed-loops with sparsity constraints, improving design efficiency over existing implicit methods.

Findings

Explicit parameterization facilitates efficient controller design.

Linear scaling of transfer function parameters with spatial constraints.

Applications demonstrated on consensus and vehicular platoons.

Abstract

In this article, we study the linear time-invariant state-feedback controller design problem for distributed systems. We follow the recently developed system level synthesis (SLS) approach and impose locality structure on the resulting closed-loop mappings; the corresponding controller implementation inherits this prescribed structure. In contrast to existing SLS results, we derive an explicit (rather than implicit) parameterization of all achievable stabilized closed-loops. This admits more efficient IIR representations of the temporal part of the closed-loop dynamics, and it allows for the H2 design problem with closed-loop spatial sparsity constraints to be converted to a standard model matching problem, with the number of transfer function parameters scaling linearly with the closed-loop spatial extent constraint. We illustrate our results with two applications: consensus of first-order subsystems and the vehicular platoons problem. In the case of first-order consensus, we provide analytic solutions and further analyze the architecture of the resulting controller implementation. Results for infinite extent spatially invariant systems are presented to provide insight to the case of a large but finite number of subsystems.